1
Distances (again)
Example 1.1.
Speedometer readings for a motorcycle at 12 second intervals are given:
t
(
s
)
0
12
24
36
48
60
v
30
28
25
22
24
27
Give two diﬀerent estimates.
2
Deﬁnite Integral
Recall what we just did.
Deﬁnition 1.
If
f
is a continuous function deﬁned for
a
≤
x
≤
b
, we divide the interval [
a, b
] into
n
subintervals of equal width Δ
x
= (
b

a
)
/n
. We let
x
0
= 1
, x
1
, . . . , x
n
=
b
be the endpoints of the intervals
and we let
x
*
1
, x
*
2
, . . . , x
*
n
be any sample points in these intervals, so
x
*
i
is in [
x
i

1
, x
i
]. The deﬁnite integral
of
f
from
a
to
b
is
Z
b
a
f
(
x
)
dx
= lim
n
→∞
n
X
i
=1
f
(
x
*
i
)Δ
x
To make things easier, we usually pick the same point in every interval, usually either the left endpoint
or the right endpoint. So usually
x
*
i
=
x
i
or
x
*
i
=
x
i

1
. Also, as long as
f
is continuous, it doesn’t matter
which one we pick. They’ll always end up the same.
Deﬁnition 2.
In
R
b
a
f
(
x
)
dx
, the
f
(
x
) is called the integrand. The
a
and
b
are limits of integration 
a
is
the lower limit and
b
is the upper limit.
dx
has no oﬃcial meaning by itself. It’s pretty much just a symbol.
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 Spring '08
 varies
 Calculus, Derivative, Fundamental Theorem Of Calculus, lim, dx

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